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CBSE Class 12 Maths Chapter 8 - Application of Integrals Revision Notes (Free PDF Download)

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Revision Notes for CBSE Class 12 Maths Chapter 8 (Application of Integrals) - Free PDF Download

Revision notes for Class 12 Maths Chapter 8 Application of Integrals include explaining all the topics provided in every chapter in a simple and easily understandable format. These revision notes help students in preparing concepts important for the exam point of view in a better way. It provides an extra edge to students and boosts their self-confidence before the exam. The concepts covered in class 12 Maths Application of Integrals are a foundational base for competitive exams like IIT JEE, Olympiad, etc.


Therefore, the students are advised to have a clear understanding of concepts. The students can prepare topics covered by downloading the Class 12 Maths Chapter 8 Application of Integrals revision notes through the link provided below:


Topics Covered In Class 12 Maths Chapter 8 Application of Integral

  • Introduction to integrals.

  • The area under simple curves.

  • The area between two curves.


Advantages of CBSE Class 12 Maths Chapter 8 Application of Integrals: Revision Notes (Free PDF Download)

The following are some benefits of using Vedantu's Class 12 Maths Chapter 8 Revision Notes.

  • It covers the key point of each topic covered in the chapter.

  • Equations are solved with a step-by-step procedure.

  • The revision notes follow the most recent CBSE syllabus requirements.

  • Referring to the revision notes is the most effective method of exam preparation.

  • It allows applicants to master all the concepts quickly, considerably reducing revision hours and allowing time for solving papers.

  • It aids in improving one's understanding of the complex concepts


Exam Preparation Tips

Following are some tips that help students perform well in the upcoming examinations.

  • Know the complete syllabus.

  • Make revision notes to get the last-minute revision.

  • Solve previous year's question papers and sample papers.

  • Always analyse the mistakes and work on them.

  • Focus on the NCERT exercises and examples as the questions are majorly asked from the NCERT book in board exams.

  • Prepare flowcharts and diagrams to explain the complicated concepts. This would help students to memorise better and also help them to retain information for a longer time. 


Applications of Integrals Extra Questions For Practice 

  1. Find the area bounded by the parabola y2 = 8x and x2 = 8y.

  2. Calculate the area of the region enclosed between the circles: x2 + y2 = 25 and (x + 5)2 + y2 = 25.

  3. Find the area between the x-axis curve y = sin x when 0≤ x < 2.

  4. Find the area bounded by the parabola x2 = 4x and the line x = 4y-2

  5. Find the region's area bounded by the curves y = x - 1 and y = 3 - x.

  6. Find the area enclosed by the curve x2y = 36. the x-axis and the lines x = 6 and x= 9.

  7. Find the area bounded by the curve y = x3, x = 0 and the ordinates x = -2 and x = 1.

  8. Calculate the area of the region bounded by the ellipse x2/4 + y2/9 = 1.

  9. Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x, and the circle x2 + y2 = 32 using the integration method.

  10. Find the area of the region bounded by the lines: 3x – 2y + 1 = 0,2x + 3y – 21 = 0 and x – 5y + 9 = 0 using integration method


CBSE Class 12 Maths Revision Notes 2024-25 - Chapter Wise PDF Notes for Free

In the table below we have provided the PDF links of all the chapters of CBSE Class 12 Maths whereby the students are required to revise the chapters by downloading the PDF. 


Application of Integrals Related Important Links

It is a curated compilation of relevant online resources that complement and expand upon the content covered in a specific chapter. Explore these links to access additional readings, explanatory videos, practice exercises, and other valuable materials that enhance your understanding of the chapter's subject matter.


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Application of Integrals Class 12 Notes Maths - Basic Subjective Questions


Section–A (1 Mark Questions)

1. Find the area of the region bounded by the curve $y=\sin x$ between the ordinates $x=0, x=\frac{\pi}{2}$ and the $x$-axis.

Ans. We have $y=\sin x, 0 \leq x \leq \frac{\pi}{2}$

From the figure, area of the shaded region


seo images


$\begin{aligned}& A=\int_0^{\pi / 2} \sin x d x=[-\cos x]_0^{\pi / 2} \\& =-\cos \frac{\pi}{2}+\cos 0=0+1=1 \text { sq. units }\end{aligned}$


2. Find the area of the region bounded by the curve $y=x+1$ and the lines $x=2$ and $x=3$.

Ans. From the figure, area of the shaded region,


seo images


$$\begin{aligned}& A=\int_2^3(x+1) d x=\left[\frac{x^2}{2}+x\right]_2^3 \\& =\left[\frac{9}{2}+3-\frac{4}{2}-2\right] \\& =\frac{7}{2} \text { sq. units }\end{aligned}$$


3. Find the area of the region bounded by the curve $x=2 y+3$ and the line $y=1$ and $y=-1$.

Ans. From the figure, area of the shaded region,


seo images


$$\begin{aligned}& A=\int_{-1}^1(2 y+3) d y=\left[y^2+3 y\right]_{-1}^1 \\& =[1+3-1+3]=6 \text { sq. units }\end{aligned}$$


4. Find the area of the region bounded by the curve $x=y^2, y$ - axis and the lines $$y=3 \text { and } y=4 .$$

Ans. $$\begin{aligned}& \text { Required area, } A=\int_3 x d y \\& =\int_3^4 y^2 d y \\& =\left[\frac{y^3}{3}\right]_3^4 \\& =\frac{1}{3}\left[4^3-3^3\right] \\& =\frac{37}{3} \text { sq. units }\end{aligned}$$


seo images


5. Using integration, find the area of the region bounded by the line $2 y=5 x+7$, $x$-axis and the lines $x=2$ and $x=8$.

Ans. We have, $2 y=5 x+7$ Or $y=\frac{5 x}{2}+\frac{7}{2}$

The graph is as shown in figure below. From the figure, area of shaded region


seo images


$$\begin{aligned}& =\int_2^8 \frac{5 x+7}{2} d x \\& =\frac{1}{2}\left[5\left(\frac{x^2}{2}\right)+7 x\right]_2^8 \\& =\frac{1}{2}[5(32)+7(8)-10-14] \\& =\frac{1}{2}[160+56-24] \\& =96 \text { sq. units. }\end{aligned}$$


Section-B (2 Marks Questions)

6. Find the area of the region bounded by the curve $y=\cos x$ between $x=0$ and $x=\pi$.

Ans. We have $y=\cos x, x=0$ and $x=\pi$

From the figure, area of the shaded region,


seo images


$$\begin{aligned}& A=\int_0^{\pi / 2} \cos x d x+\int_{\pi / 2}^\pi|\cos x| d x \\& =2 \int_0^{\frac{\pi}{2}} \cos x d x=2[\sin x]_0^{\pi / 2}=2 \text { sq. units. }\end{aligned}$$


7. Find the area bounded by the parabola $y^2=4 x$ and line $x=3$.

Ans. Required Area $=$ area $\mathrm{AOBMA}=2 \times \operatorname{area}($ AOMA $)$ ($\because$ Parabola is symmetrical about $x$-axis)


seo images


$$\begin{aligned}& =2 \int_0^a y d x=2 \int_0^a \sqrt{4 a x} d x \\& =2 \times 2 \sqrt{a} \int_0^a \sqrt{x} d x=4 \sqrt{a} \times\left[\frac{2}{3}x^{\frac{3}{2}}\right]_0^a \\& =\frac{8 \sqrt{a}}{3}\left[a^{3 / 2}\right]=\frac{8}{3} a^2 \quad \text { sq. units. }\end{aligned}$$sq. units. I


8. Find the area of the region bounded by the parabolas $y^2=6 x$ and $x^2=6 y$.

Ans. The intersecting points of the given parabolas are obtained by solving these equations for $x$ and $y$, which are $O(0,0)$ and $(6,6)$.


seo images


$$\begin{aligned}& \text { Area } O A B C O=\int_0^6\left(\sqrt{6 x}-\frac{x^2}{6}\right) d x \\& =\left[2 \sqrt{6} \frac{x^{\frac{3}{2}}}{3}-\frac{x^3}{18}\right]_0^6 \\& =2 \sqrt{6} \frac{(6)^{\frac{3}{2}}}{3}-\frac{(6)^3}{18}=12 \text { sq. units. }\end{aligned}$$


9. The area of the region bounded by the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is

Ans. We have, $\frac{x^2}{25}+\frac{y^2}{16}=1$ which is ellipse with its axis as coordinate axis.


$$\begin{aligned}& \frac{y^2}{16}=1-\frac{x^2}{25} \Rightarrow y^2=16\left(1-\frac{x^2}{25}\right) \\& \Rightarrow y=\frac{4}{5} \sqrt{5^2-x^2}\end{aligned}$$


From the figure, area of the shaded region.


seo images


$$\begin{aligned}& A=4 \int_0^5 \frac{4}{5} \sqrt{5^2-x^2} d x \\& =\frac{16}{5}\left[\frac{x}{2} \sqrt{5^2-x^2}+\frac{5^2}{2} \sin^{-1}\left(\frac{x}{5}\right)\right]_0^5\end{aligned}$$


$$\begin{aligned}& =\frac{16}{5}\left[0+\frac{5^2}{2} \sin ^{-1}(1)-0-0\right] \\& =\frac{16}{5} \cdot \frac{25}{2} \cdot \frac{\pi}{2}=20 \pi \text { sq. units. }\end{aligned}$$


10. Find the area of the parabola $y^2=4 a x$ bounded by its latus rectum.

Ans. The vertex of the parabola $y^2=4 a x$ is at origin $(0,0)$. The equation of the latus rectum LSL' is $x=a$. Also, parabola is symmetrical about the x-axis.

The required area of the region $O L L^{\prime} \mathrm{O}$ $=2 \times($ Area of the region $\mathrm{O} L S O)$.


seo images


$=2 \int_0^a y d x=2 \int_0^a \sqrt{4 a x} d x$

$=2 \times 2 \sqrt{a} \int_0^a \sqrt{x} d x=4 \sqrt{a} \times\left[\frac{2}{3}x^{\frac{3}{2}}\right]_0^a$

$=\frac{8 \sqrt{a}}{3}\left[a^{3 / 2}\right]=\frac{8}{3} a^2 \quad \text { sq. units. }$


11. Find the area enclosed by parabola $y=4 x^2$ and lines, $y=1$ and $y=4$.

Ans. Parabola $y=4 x^2$

So $x^2=\frac{1}{4} y$ and lines $y=1$ and $y=4$ will be traced as followed.

So, required area $=$ area $P Q R S P$


seo images


$$\begin{aligned}& =2 \times \operatorname{Area}(\text { RQLMR }) \\& =2 \int_1^4 x d y=2 \int_1^4 \frac{1}{2} \sqrt{y} d y=\int_1^4 \sqrt{y} d y \\& =\frac{2}{3}\left[(y)^{3 / 2}\right]_1^4=\frac{2}{3}\left[4^{3 / 2}-1^{3 / 2}\right] \\& =\frac{2}{3}[8-1]=\frac{14}{3} \text { sq. units. }\end{aligned}$$


12. Find the area enclosed by the curve $y^{y=-x^2}$ and the straight line $x+y+2=0$.

Ans. We have, $y=-x^2$ and $x+y+2=0$

Solving we get, $$\begin{aligned}& x^2=x+2 \\& \Rightarrow x^2-x-2=0 \\& \Rightarrow(x-2)(x+1)=0 \\& \Rightarrow x=2,-1\end{aligned}$$


The graph of above function is downward parabola. From the figure, area of shaded region,


seo images


$A=\int_{-1}^2\left(-x^2-(-x-2)\right) d x$

$=\int_{-1}^2\left(x+2-x^2\right) d x$

$=\left[\frac{x^2}{2}+2 x-\frac{x^3}{3}\right]_{-1}^2$

$=\left[\frac{2^2}{2}+2(2)-\frac{2^3}{3}\right]-\left [ \frac{(-1)^{2}}{2}+(-1)-\frac{(-1)^{3}}{3} \right ]$

$=\left[6-\frac{8}{3}\right]-\left[\frac{1}{2}-2+\frac{1}{3}\right]$

$=\frac{9}{2} \text { sq. units }$


13. Find the area bounded by the curve $y=\sqrt{x}, \quad x=2 y+3$ in the first quadrant and $x$-axis.

Ans. We have $y=\sqrt{x}$ and $x=2 y+3$

Solving we get


$$ \begin{aligned} & y=\sqrt{2 y+3}, y \geq 0 \\ & \Rightarrow y^2=2 y+3, y \geq 0 \\ & \Rightarrow y^2-2 y-3=0, y \geq 0 \\ & \Rightarrow(y-3)(y+1)=0, y \geq 0 \\ & \Rightarrow y=3 \end{aligned} $$


The graph of function $y=\sqrt{x}$ is part of parabola $y^2=x$ lying above $x$-axis. The graph is as shown in the following figure.

From the figure, area of shaded region,


seo images


$$ \begin{aligned} & A=\int_0^3\left(2 y+3-y^2\right) d y \\ & =\left[y^2+3 y-\frac{y^3}{3}\right]_0^3 \\ & =[9+9-9-0] \\ & =9 \text { sq. units. } \end{aligned} $$


PDF Summary - Class 12 Maths Application of Integrals Notes (Chapter 8)

Definite Integration 

1. Definition:

If \[\text{ }\!\!~\!\!\text{ F}\left( \text{x} \right)\] is an antiderivative of \[\text{f}\left( \text{x} \right)\text{,}\] then \[\text{ }\!\!~\!\!\text{ F}\left( \text{b} \right)\text{--F}\left( \text{a} \right)\] is known as the definite integral of \[\text{f}\left( \text{x} \right)\] from $\text{a}$ to $\text{b}$, such that the variable \[\text{x,}\] takes any two independent  values say \[\text{a}\] and \[\text{b}\].

This is also denoted as  $\int_{\text{a}}^{\text{b}}{\text{f(x)dx}}$. 

Thus $\int_{\text{a}}^{\text{b}}{\text{f(x)dx}}=\text{F}\left( \text{b} \right)-\text{F}\left( \text{a} \right)$, The numbers $\text{a}$ and $\text{b}$ are called the limits of integration; $\text{a}$ is the lower limit and $\text{b}$ is the upper limit. Usually $\text{F}\left( \text{b} \right)-\text{F}\left( \text{a} \right)$ is abbreviated by writing $\text{F}\left( \text{x} \right)\left| _{\text{a}}^{\text{b}} \right.$.


2. Properties of Definite Integrals:

l. $\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx=}-\int\limits_{\text{b}}^{\text{a}}{\text{f(x)}}}$ 

ll. $\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx=}\int\limits_{\text{b}}^{\text{a}}{\text{f(y)}}}\text{dy}$ 

lll. $\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx=}\int\limits_{\text{a}}^{\text{c}}{\text{f(x)dx+}\int\limits_{\text{c}}^{\text{b}}{\text{f(x)dx}}}}$, where $\text{c}$ may or may not lie between $\text{a}$ and $\text{b}$.

lV. $\int\limits_{\text{0}}^{\text{a}}{\text{f(x)dx=}\int\limits_{\text{0}}^{\text{a}}{\text{f(a}-\text{x)dx}}}$ 

V. $\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx=}\int\limits_{\text{a}}^{\text{b}}{\text{f(a+b}-\text{x)dx}}}$ 

Note:

  1. $\int\limits_{\text{0}}^{\text{a}}{\dfrac{\text{f(x)}}{\text{f(x)+f(a}-\text{x)}}}\text{dx=}\dfrac{\text{a}}{\text{2}}$ 

  2. $\int\limits_{\text{a}}^{\text{b}}{\dfrac{\text{f(x)}}{\text{f(x)+f(a+b}-\text{x)}}}\text{dx=}\dfrac{\text{b}-\text{a}}{\text{2}}$ 

Vl. $\int\limits_{\text{0}}^{\text{2a}}{\text{f(x)dx=}\int\limits_{\text{0}}^{\text{a}}{\text{f(x)dx+}\int\limits_{\text{0}}^{\text{a}}{\text{f(2a}-\text{x)dx}}}}$ 

$\text{=}\left\{ \begin{matrix} \text{0}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\text{f(2a}-\text{x)}=-\text{f(x)}  \\ \text{2}\int\limits_{\text{0}}^{\text{a}}{\text{f(x)dx}\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\text{f(2a}-\text{x)}=\text{f(x)}}  \\ \end{matrix} \right\}$ 

Vll. \[\int\limits_{\text{-a}}^{\text{a}}{\text{f(x)dx}=\left\{ \begin{matrix} \text{2}\int\limits_{\text{0}}^{\text{a}}{\text{f(x)dx}\,\,\,\,\,\,\,\,\,\,\text{if}\,\text{f(}-\text{x)}=\text{f(x)}\,\text{i}\text{.e}\text{.}\,\text{f(x)}\,\text{is}\,\text{even}}  \\ \text{0}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if}\,\text{f(}-\text{x)}=-\text{f(x)}\,\text{i}\text{.e}\text{.}\,\text{f(x)}\,\text{is}\,\text{odd}  \\ \end{matrix} \right\}}\] 

Vlll. If $\text{f(x)}$ is a periodic function of period $\text{ }\!\!'\!\!\text{ a }\!\!'\!\!\text{ }$, i.e. $\text{f(a}+\text{x)}=\text{f(x)}$, then

  1. $\int\limits_{\text{0}}^{\text{na}}{\text{f(x)dx}=\text{n}\int\limits_{\text{0}}^{\text{a}}{\text{f(x)dx}}}$ 

  2. $\int\limits_{\text{0}}^{\text{na}}{\text{f(x)dx}=\left( \text{n}-1 \right)\int\limits_{\text{0}}^{\text{a}}{\text{f(x)dx}}}$ 

  3. $\int\limits_{\text{na}}^{\text{b+na}}{\text{f(x)dx}=\int\limits_{\text{0}}^{\text{b}}{\text{f(x)dx}}}$, where $\text{b}\in \text{R}$ 

  4. $\int\limits_{\text{b}}^{\text{b+a}}{\text{f(x)dx}}$ independent of $\text{b}$.

  5. $\int\limits_{\text{b}}^{\text{b+na}}{\text{f(x)dx}=\text{n}\int\limits_{\text{0}}^{\text{a}}{\text{f(x)dx}}}$, where $\text{n}\in \text{1}$ 

lX. If $\text{f(x)}\ge \text{0}$ on the interval $\left[ \text{a,b} \right]$, then $\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx}}\ge 0$.

X. If $\text{f(x)}\le \text{g(x)}$ on the interval $\left[ \text{a,b} \right]$, then $\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx}}\le \int\limits_{\text{a}}^{\text{b}}{\text{g(x)dx}}$ 

X. $\left| \int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx}} \right|\le \int\limits_{\text{a}}^{\text{b}}{\left| \text{f(x)} \right|\text{dx}}$ 

Xl.  If $\text{f(x)}$ is continuous on $\left[ \text{a,b} \right]$, $\text{m}$ is the least and $\text{M}$ is the greatest value of $\text{f(x)}$ on $\left[ \text{a,b} \right]$, then

$\text{m}\left( \text{b}-\text{a} \right)\le \int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx}}\le \text{M(b}-\text{a)}$ 

Xll. For any two functions $\text{f(x)}$ and $\text{g(x)}$, integral on the interval $\left[ \text{a,b} \right]$, the Schwarz- Bunyakovsky inequality holds

$\left| \int\limits_{\text{a}}^{\text{b}}{\text{f(x)}\text{.g(x)dx}} \right|\le \sqrt{\int\limits_{\text{a}}^{\text{b}}{{{\text{f}}^{\text{2}}}\text{(x)dx}\text{.}\int\limits_{\text{a}}^{\text{b}}{{{\text{g}}^{\text{2}}}\text{(x)dx}}}}$ 

Xlll. If a function $\text{f(x)}$ is continuous on the interval $\left[ \text{a,b} \right]$, then there exists a point $\text{c}\in \left( \text{a,b} \right)$ such that

\[\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx}=\text{f(c)(b}-\text{a)}}\], where $\text{a}<\text{c}<\text{b}$.


3. Differentiation Under Integral Sign:

Newton Leibnitz’s Theorem:

Given that $\text{x}$ has two differentiable functions, $\text{g(x)}$ and $\text{h(x)}$, where $\text{x}\in \left[ \text{a,b} \right]$ and $\text{f}$ is continuous in that interval, then  

$\dfrac{\text{d}}{\text{dx}}\left[ \int\limits_{\text{g(x)}}^{\text{h(x)}}{\text{f(t)dt}} \right]=\dfrac{\text{d}}{\text{dx}}\left[ \text{h(x)} \right]\text{.f}\left[ \text{h(x)} \right]-\dfrac{\text{d}}{\text{dx}}\left[ \text{g(x)} \right]\text{.f}\left[ \text{g(x)} \right]$ 


4. Definite Integral as a Limit of Sum:

Let $\text{f(x)}$ be a continuous real valued function defined on the closed interval $\left[ \text{a,b} \right]$ which is divided into $\text{n}$ parts as shown in figure.


Definite Integral as a Limit of Sum

The point of division on $\text{x-}$axis are

$\text{a,a}+\text{h,a}+\text{2h}.....\text{a}+\left( \text{n}-\text{1} \right)\text{h,a}+\text{nh,}$ where $\dfrac{\text{b}-\text{a}}{\text{n}}=\text{h}$.

Let ${{\text{S}}_{\text{n}}}$ denotes the area of these $\text{n}$ rectangles.

Then, ${{\text{S}}_{\text{n}}}=\text{hf(a)}+\text{hf}\left( \text{a}+\text{h} \right)+\text{hf}\left( \text{a}+\text{2h} \right)+....+\text{hf}\left( \text{a}+\left( \text{n}-\text{1} \right)\text{h} \right)$ 

Clearly, ${{\text{S}}_{\text{n}}}$ is area very close to the area of the region bounded by

Curve $\text{y}=\text{f(x),}$ $\text{x-}$axis and the ordinates $\text{x}=\text{a}$, $\text{x}=\text{b}$.

Hence \[\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx}=\underset{\text{n}\to \infty }{\mathop{\text{Lt}}}\,}{{\text{S}}_{\text{n}}}\] 

\[\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx}=\underset{\text{n}\to \infty }{\mathop{\text{Lt}}}\,}\sum\limits_{\text{r}=\text{0}}^{\text{n-1}}{\text{hf(a+rh)}}\] 

\[=\underset{\text{n}\to \infty }{\mathop{\text{Lt}}}\,\sum\limits_{\text{r}=\text{0}}^{\text{n-1}}{\left( \dfrac{\text{b}-\text{a}}{\text{n}} \right)}\text{f}\left( \text{a}+\dfrac{\left( \text{b}-\text{a} \right)\text{r}}{\text{n}} \right)\] 

Note:

(a) We can also write

${{\text{S}}_{\text{n}}}=\text{hf}\left( \text{a}+\text{h} \right)+\text{hf}\left( \text{a}+\text{2h} \right)+.....+\text{hf}\left( \text{a}+\text{nh} \right)$ 

\[\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx}}=\underset{\text{n}\to \infty }{\mathop{\text{Lt}}}\,\sum\limits_{\text{r}=1}^{\text{n}}{\left( \dfrac{\text{b}-\text{a}}{\text{n}} \right)}\text{f}\left( \text{a}+\left( \dfrac{\text{b}-\text{a}}{\text{n}} \right)\text{r} \right)\] 

(b) If \[\text{a}=0,\,\text{b}=1,\,\int\limits_{0}^{1}{\text{f(x)dx}}=\underset{\text{n}\to \infty }{\mathop{\text{Lt}}}\,\sum\limits_{\text{r}=0}^{\text{n}-1}{\dfrac{1}{\text{n}}\text{f}\left( \dfrac{\text{r}}{\text{n}} \right)}\] 


Steps to Express the Limit of Sum as Definite Integral

Step 1. Replace  \[\dfrac{\text{r}}{\text{n}}\] by $\text{x}$, $\dfrac{\text{1}}{\text{n}}$ by $\text{dx}$ and $\underset{\text{n}\to \infty }{\mathop{\text{Lt}}}\,\sum{\text{by}\,}\int{{}}$  

Step 2. Evaluate $\underset{\text{n}\to \infty }{\mathop{\text{Lt}}}\,\left( \dfrac{\text{r}}{\text{n}} \right)$ by putting least and greatest values of $\text{r}$ as lower and upper limits respectively.

For example $\underset{\text{n}\to \infty }{\mathop{\text{Lt}}}\,\sum\limits_{\text{r}=1}^{\text{pn}}{\dfrac{1}{\text{n}}\text{f}\left( \dfrac{\text{r}}{\text{n}} \right)}=\int\limits_{0}{\text{f(x)dx}}$ 

$\left[ \underset{\text{n}\to \infty }{\mathop{\text{Lt}}}\,\left( \dfrac{\text{r}}{\text{n}} \right)\left| _{\text{r}=1}=0,\,\underset{\text{n}\to \infty }{\mathop{\text{Lt}}}\, \right.\left( \dfrac{\text{r}}{\text{n}} \right)\left| _{\text{r}=\text{np}}=\text{p} \right. \right]$ 


5. Reduction Formulae in Definite Integrals

l. If ${{\text{I}}_{\text{n}}}\text{=}\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{si}{{\text{n}}^{\text{n}}}\text{xdx}}$, then show that ${{\text{I}}_{\text{n}}}\text{=}\left( \dfrac{\text{n}-\text{1}}{\text{n}} \right){{\text{I}}_{\text{n}-\text{2}}}$ 

Proof: ${{\text{I}}_{\text{n}}}\text{=}\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{si}{{\text{n}}^{\text{n}}}\text{xdx}}$ 

${{\text{I}}_{\text{n}}}\text{=}\left[ -\text{si}{{\text{n}}^{\text{n}-\text{1}}}\text{xcosx} \right]_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}\text{+}\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\left( \text{n}-\text{1} \right)\text{si}{{\text{n}}^{\text{n}-\text{2}}}\text{x}\text{.co}{{\text{s}}^{\text{2}}}\text{xdx}}$ 

$\text{=}\left( \text{n}-\text{1} \right)\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{si}{{\text{n}}^{\text{n}-\text{2}}}\text{x}\text{.}\left( \text{1}-\text{si}{{\text{n}}^{\text{2}}}\text{x} \right)\text{dx}}$ 

$\text{=}\left( \text{n}-\text{1} \right)\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{si}{{\text{n}}^{\text{n}-\text{2}}}\text{xdx}-\left( \text{n}-1 \right)\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{si}{{\text{n}}^{\text{n}}}\text{x}}\text{dx}}$ 

${{\text{I}}_{\text{n}}}+\left( \text{n}-\text{1} \right){{\text{I}}_{\text{n}}}=\left( \text{n}-\text{1} \right){{\text{I}}_{\text{n}-\text{2}}}$ 

${{\text{I}}_{\text{n}}}=\left( \dfrac{\text{n}-\text{1}}{\text{n}} \right){{\text{I}}_{\text{n}-\text{2}}}$ 

Note:

(a) $\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{si}{{\text{n}}^{\text{n}}}\text{xdx}=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{co}{{\text{s}}^{\text{n}}}\text{xdx}}}$ 

(b) ${{\text{I}}_{\text{n}}}=\left( \dfrac{\text{n}-\text{1}}{\text{n}} \right)\left( \dfrac{\text{n}-\text{3}}{\text{n}-\text{2}} \right)\left( \dfrac{\text{n}-\text{5}}{\text{n}-\text{4}} \right)....{{\text{I}}_{\text{0}}}\,\text{or}\,{{\text{I}}_{\text{1}}}$ according as \[\text{n}\] is even or odd, ${{\text{I}}_{\text{0}}}=\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}\text{,}\,{{\text{I}}_{\text{1}}}=\text{1}$ 

Hence ${{\text{I}}_{\text{n}}}=\left\{ \begin{matrix} \left( \dfrac{\text{n}-\text{1}}{\text{n}} \right)\left( \dfrac{\text{n}-\text{3}}{\text{n}-\text{2}} \right)\left( \dfrac{\text{n}-\text{5}}{\text{n}-\text{4}} \right).....\left( \dfrac{\text{1}}{\text{2}} \right)\text{.}\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}\,\,\text{if}\,\,\text{n}\,\,\text{is}\,\,\text{even}  \\ \left( \dfrac{\text{n}-\text{1}}{\text{n}} \right)\left( \dfrac{\text{n}-\text{3}}{\text{n}-\text{2}} \right)\left( \dfrac{\text{n}-\text{5}}{\text{n}-\text{4}} \right).....\left( \dfrac{\text{2}}{\text{3}} \right)\text{.1}\,\,\text{if}\,\,\text{n}\,\,\text{is }\,\text{odd}  \\ \end{matrix} \right\}$ 

ll. If ${{\text{I}}_{\text{n}}}=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{\text{ta}{{\text{n}}^{\text{n}}}\text{xdx}}$, then show that ${{\text{I}}_{\text{n}}}+{{\text{I}}_{\text{n}-\text{2}}}=\dfrac{\text{1}}{\text{n}-\text{1}}$ 

Proof: ${{\text{I}}_{\text{n}}}=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{{{\left( \text{tanx} \right)}^{\text{n}-\text{2}}}\text{.ta}{{\text{n}}^{\text{2}}}\text{xdx}}$ 

$=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{{{\left( \text{tanx} \right)}^{\text{n}-\text{2}}}\left( \text{se}{{\text{c}}^{\text{2}}}\text{x}-\text{1} \right)\text{dx}}$ 

$\text{=}\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{{{\left( \text{tanx} \right)}^{\text{n}-\text{2}}}\text{se}{{\text{c}}^{\text{2}}}\text{xdx}}-\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{{{\left( \text{tanx} \right)}^{\text{n}-\text{2}}}\text{dx}}$ 

$=\left[ \dfrac{{{\left( \text{tanx} \right)}^{\text{n}-\text{1}}}}{\text{n}-\text{1}} \right]_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}-{{\text{I}}_{\text{n}-\text{2}}}$ 

${{\text{I}}_{\text{n}}}=\dfrac{\text{1}}{\text{n}-\text{1}}-{{\text{I}}_{\text{n}-\text{2}}}$ 

${{\text{I}}_{\text{n}}}+{{\text{I}}_{\text{n}-\text{2}}}=\dfrac{\text{1}}{\text{n}-\text{1}}$ 

lll. If ${{\text{I}}_{\text{m,n}}}=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{si}{{\text{n}}^{\text{m}}}\text{x}\text{.co}{{\text{s}}^{\text{n}}}\text{xdx}}$, then show that ${{\text{I}}_{\text{m,n}}}=\dfrac{\text{m}-\text{1}}{\text{m}+\text{n}}{{\text{I}}_{\text{m}-\text{2}}}\text{,n}$ 

Proof: ${{\text{I}}_{\text{m,n}}}=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{si}{{\text{n}}^{\text{m}-1}}\text{x}\left( \text{sinxco}{{\text{s}}^{\text{n}}}\text{x} \right)\text{dx}}$ 

$=\left[ -\dfrac{\text{si}{{\text{n}}^{\text{m}-\text{1}}}\text{x}\text{.co}{{\text{s}}^{\text{n}+\text{1}}}\text{x}}{\text{n}+\text{1}} \right]_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}+\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\dfrac{\text{co}{{\text{s}}^{\text{n}+\text{1}}}}{\text{n}+\text{1}}\left( \text{m}-\text{1} \right)\text{si}{{\text{n}}^{\text{m}-\text{2}}}\text{xcosxdx}}$ 

$=\left( \dfrac{\text{m}-\text{1}}{\text{n}+\text{1}} \right)\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{si}{{\text{n}}^{\text{m}-\text{2}}}\text{x}\text{.co}{{\text{s}}^{\text{n}}}\text{x}\text{.co}{{\text{s}}^{\text{2}}}\text{xdx}}$ 

$=\left( \dfrac{\text{m}-\text{1}}{\text{n}+\text{1}} \right)\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\left( \text{si}{{\text{n}}^{\text{m}-\text{2}}}\text{x}\text{.co}{{\text{s}}^{\text{n}}}\text{x}-\text{si}{{\text{n}}^{\text{m}}}\text{x}\text{.co}{{\text{s}}^{\text{n}}}\text{x} \right)\text{dx}}$ 

$=\left( \dfrac{\text{m}-\text{1}}{\text{n}+\text{1}} \right){{\text{I}}_{\text{m}-\text{2,n}}}-\left( \dfrac{\text{m}-\text{1}}{\text{n}+\text{1}} \right){{\text{I}}_{\text{m,n}}}$ 

$\Rightarrow \left( \text{1}+\dfrac{\text{m}-\text{1}}{\text{n}+\text{1}} \right){{\text{I}}_{\text{m,n}}}=\left( \dfrac{\text{m}-\text{1}}{\text{n}+\text{1}} \right){{\text{I}}_{\text{m}-\text{2,n}}}$ 

${{\text{I}}_{\text{m,n}}}\text{=}\left( \dfrac{\text{m}-\text{1}}{\text{m}+\text{n}} \right){{\text{I}}_{\text{m}-\text{2,n}}}$ 

Note:

(a) ${{\text{I}}_{\text{m,n}}}=\left( \dfrac{\text{m}-\text{1}}{\text{m}+\text{n}} \right)\left( \dfrac{\text{m}-\text{3}}{\text{m}+\text{n}-\text{2}} \right)\left( \dfrac{\text{m}-\text{5}}{\text{m}+\text{n}-\text{4}} \right).....{{\text{I}}_{\text{0,n}}}\,\text{or}\,{{\text{I}}_{\text{1,n}}}$ according as $\text{m}$ is even or odd.

${{\text{I}}_{\text{0,n}}}=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{co}{{\text{s}}^{\text{n}}}\text{xdx}\,\text{and}\,{{\text{I}}_{\text{1,n}}}=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\text{sinx}\text{.co}{{\text{s}}^{\text{n}}}\text{xdx}=\dfrac{\text{1}}{\text{n+1}}}}$ 

(b) Walli’s Formula

\[{{\text{I}}_{\text{m,n}}}=\left\{ \begin{matrix} \dfrac{\left( \text{m}-\text{1} \right)\left( \text{m}-\text{3} \right)\left( \text{m}-\text{5} \right)....\left( \text{n}-\text{1} \right)\left( \text{n}-\text{3} \right)\left( \text{n}-\text{5} \right)....}{\left( \text{m}+\text{n} \right)\left( \text{m}+\text{n}-\text{2} \right)\left( \text{m}+\text{n}-\text{4} \right).....}\,\,\,\text{when}\,\,\text{both}\,\,\text{m,n}\,\,\text{are}\,\,\text{even}  \\ \dfrac{\left( \text{m}-\text{1} \right)\left( \text{m}-\text{3} \right)\left( \text{m}-\text{5} \right)....\left( \text{n}-\text{1} \right)\left( \text{n}-\text{3} \right)\left( \text{n}-\text{5} \right).....}{\left( \text{m}+\text{n} \right)\left( \text{m}+\text{n}-\text{2} \right)\left( \text{m}+\text{n}-\text{4} \right)....}\,\,\text{otherwise}  \\ \end{matrix} \right\}\] 


Area Under the Curves

(a) Area of Plane Regions:

l. The area bounded by the curve $\text{y}=\text{f(x)}$, $\text{x-}$axis and the ordinates $\text{x}=\text{a}$ and \[\text{x}=\text{b}\] (where $\text{ba}$) is given by

$\text{A}=\int\limits_{\text{a}}^{\text{b}}{\left| \text{y} \right|\text{dx}=\int\limits_{\text{a}}^{\text{b}}{\left| \text{f(x)} \right|\text{dx}}}$ 

(i) If $\text{f(x)}>0\forall \text{x}\in \left[ \text{a,b} \right]$ 

Then $\text{A}=\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx}}$ 


Area of plane regions

(ii) If $\text{f(x)}>0\forall x\in \left[ \text{a,c} \right)\,\text{ }\!\!\And\!\!\text{ }\,<0\forall \text{x}\in \left( \text{c,b} \right]$ 

Then $\text{A}=\left| \int\limits_{\text{a}}^{\text{c}}{\text{ydx}} \right|+\left| \int\limits_{\text{c}}^{\text{b}}{\text{ydx}} \right|=\int\limits_{\text{a}}^{\text{c}}{\text{f(x)dx}-\int\limits_{\text{c}}^{\text{b}}{\text{f(x)dx}}}$ where $\text{c}$ is a point in between $\text{a}$ and $\text{b}$.


Area between x=a and x=b

ll. The area bounded by the curve $\text{x}=\text{g(y)}$, $\text{y-}$axis and the abscissae $\text{y}=\text{c}$ and $\text{y}=\text{d}$ (where $\text{dc}$) is given by

$\text{A}=\int\limits_{\text{c}}^{\text{d}}{\left| \text{x} \right|\text{dy}=\int\limits_{\text{c}}^{\text{d}}{\left| \text{g(y)} \right|\text{dy}}}$ 


Area of the curve x = g(y)

III. If we have two curve $\text{y}=\text{f(x)}$ and $\text{y}=\text{g(x)}$, such that $\text{y}=\text{f(x)}$ lies above the curve $\text{y}=\text{g(x)}$ then the area bounded between them and the ordinates $\text{x}=\text{a}$ and $\text{x}=\text{b}$ $\left( \text{b}>\text{a} \right)$, is given by

$\text{A}=\int\limits_{\text{a}}^{\text{b}}{\text{f(x)dx}-\int\limits_{\text{a}}^{\text{b}}{\text{g(x)dx}}}$ 

i.e. $\text{upper}\,\text{curve}\,\text{area}-\text{lower}\,\text{curve}\,\text{area}$.


Area of two curves y=f(x) and y=g(x) with x=a and x=b

lV. The area bounded by the curves $\text{y}=\text{f(x)}$ and $\text{y}=\text{g(x)}$ between the ordinates $\text{x}=\text{a}$ and $\text{x}=\text{b}$ is given by

$\text{A}=\int\limits_{\text{a}}^{\text{c}}{\text{f(x)dx}+\int\limits_{\text{c}}^{\text{b}}{\text{g(x)dx}}}$, where $\text{x}=\text{c}$ is the point of intersection of the two curves.


Area of two curves y=f(x) and y=g(x) with x=a

V. Curve Tracing

It is necessary to have a rough sketch of the desired piece in order to locate the area enclosed by many curves. The following steps are very useful in tracing a cartesian curve $\text{f(x,y)}=\text{0}$.


Step 1: Symmetry

(i) If all the powers of $\text{y}$ in the equation of the given curve are even then the curve is symmetrical about $\text{x-}$axis.

(ii) If all the powers of $\text{x}$ in the equation of the given curve are even then the curve is symmetrical about $\text{y-}$axis.

(iii) If the equation of the given curve remains unchanged on interchanging $\text{x}$ and $\text{y}$, then the curve is symmetrical about the line\[\text{y}=\text{x}\].

(iv) If the equation of the given curve remains unchanged when $\text{x}$  and $\text{y}$ are replaced by $-\text{x}$  and $-\text{y}$ respectively, then the curve is symmetrical in opposite quadrants.


Step 2: Origin

If the algebraic curve's equation contains no constant term, the curve passes through the origin.

The tangents at the origin are then calculated by equating the lowest degree terms in the equation of the specified algebraic curve to zero.

For example, the curve ${{\text{y}}^{\text{3}}}={{\text{x}}^{\text{3}}}+\text{axy}$ passes through the origin and the tangents at the origin are given by $\text{axy}=0$ i.e. $\text{x}=\text{0}$ and $\text{y}=0$.


Step 3: Intersection with the Co-ordinates Axes

(i) Find out the corresponding values of $\text{x}$ by putting \[\text{y}=\text{0}\] in the equation of the given curve, to estimate the points of intersection of the curve with $\text{x-}$axis

(ii) Find out the corresponding values of $\text{y}$ by putting \[\text{x}=\text{0}\] in the equation of the given curve, to estimate the points of intersection of the curve with $\text{y-}$axis


Step 4: Asymptotes

Find out the asymptotes of the curve.

(i) The vertical asymptotes of the given algebraic curve, or asymptotes parallel to the $\text{y-}$axis, are derived by equating the coefficient of the highest power of $\text{y}$ in the equation of the supplied curve to zero.

(ii) The horizontal asymptotes of the given algebraic curve, or asymptotes parallel to the $\text{x-}$axis, are derived by equating the coefficient of the highest power of $\text{x}$ in the equation of the supplied curve to zero.


Step 5: Region

Find out the regions of the plane in which no part of the curve lies. To determine such regions we solve the given equation for $\text{y}$ in terms of $\text{x}$ or vice-versa. Suppose that y becomes imaginary for\[\text{x}>\text{a}\], the curve does not lie in the region\[\text{x}>\text{a}\].


Step 6: Critical Points

Find out the values of $\text{x}$ at which $\dfrac{\text{dy}}{\text{dx}}=0$ 

At such points $\text{y}$ generally changes its character from an increasing function of $\text{x}$to a decreasing function of $\text{x}$ or vice-versa.


Step 7: 

Trace the curve with the help of the above points.


Solved Examples

Definite Integration

Example - 1

Evaluate the Following Integrals:

(i) $\int\limits_{\text{2}}^{\text{3}}{{{\text{x}}^{\text{2}}}\text{dx}}$ 

(ii) $\int\limits_{\text{1}}^{\text{3}}{\dfrac{\text{x}}{\left( \text{x}+\text{1} \right)\left( \text{x}+\text{2} \right)}}\text{dx}$ 

Ans: 

(i) $\int\limits_{\text{2}}^{\text{3}}{{{\text{x}}^{\text{2}}}\text{dx}}$ 

$=\left[ \dfrac{{{\text{x}}^{\text{3}}}}{\text{3}} \right]_{\text{2}}^{\text{3}}$ 

$=\dfrac{27}{3}-\dfrac{8}{3}$ 

$=\dfrac{19}{3}$ 

(ii) $\dfrac{\text{x}}{\left( \text{x}+\text{1} \right)\left( \text{x}+\text{2} \right)}=\dfrac{-\text{1}}{\text{x}+\text{1}}+\dfrac{\text{2}}{\text{x}+\text{2}}$                [Partial Fractions]

$\int\limits_{\text{1}}^{\text{3}}{\dfrac{\text{x}}{\left( \text{x}+\text{1} \right)\left( \text{x}+\text{2} \right)}\text{dx}}$ 

$=\left[ -\log \left| \text{x}+1 \right|+2\log \left| \text{x}+2 \right| \right]_{1}^{3}$ 

$=\left[ -\log \left| 4 \right|+2\log \left| 5 \right| \right]-\left[ -\log \left| 2 \right|+2\log \left| 3 \right| \right]$ 

$=\left[ -\log 4+2\log 5 \right]-\left[ -\log 2+2\log 3 \right]$ 

$=-2\log 2+2\log 5+\log 2-2\log 3$ 

$=-\log 2+\log 25-\log 9=\log 25-\log 18$ 

$=\log \dfrac{25}{18}$ 


Example - 2

Evaluate: $\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{\text{secx}\text{.}\sqrt{\dfrac{\text{1}-\text{sinx}}{\text{1}+\text{sinx}}}}\text{dx}$ 

Ans:

$\text{I}=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{\text{secx}\text{.}\sqrt{\dfrac{\text{1}-\text{sinx}}{\text{1}+\text{sinx}}}}\text{dx}$ 

$=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{\text{secx}\text{.}\sqrt{\dfrac{\text{1}-\text{sinx}}{\text{1}+\text{sinx}}}.\sqrt{\dfrac{\text{1}-\text{sinx}}{\text{1}+\text{sinx}}}}\text{dx}$ 

$=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{\text{secx}\dfrac{\text{1}-\text{sinx}}{\sqrt{\text{1}-\text{si}{{\text{n}}^{\text{2}}}\text{x}}}}\text{dx}$ 

$=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{\text{secx}\dfrac{\text{1}-\text{sinx}}{\text{cosx}}}\text{dx}$ 

$=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{\left( \text{se}{{\text{c}}^{\text{2}}}\text{x}-\text{secxtanx} \right)\text{dx}}$ 

$=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{\text{se}{{\text{c}}^{\text{2}}}\text{xdx}-\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}{\text{secxtanxdx}}}$ 

$=\left[ \text{tanx} \right]_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}-\left[ \text{secx} \right]_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}}$ 

$=\left( \text{tan}\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}-\text{tan0} \right)-\left( \text{sec}\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}}-\text{sec0} \right)$ 

$=\left( 1-0 \right)-\left( \sqrt{2}-1 \right)=2-\sqrt{2}$.


Example - 3

Evaluate: $\int\limits_{\text{-1}}^{\text{1}}{\text{5}{{\text{x}}^{\text{4}}}\sqrt{{{\text{x}}^{\text{5}}}\text{+1}}\text{dx}}$ 

Ans:

Let $\text{I}=\int\limits_{\text{-1}}^{\text{1}}{\text{5}{{\text{x}}^{\text{4}}}\sqrt{{{\text{x}}^{\text{5}}}\text{+1}}\text{dx}}$ 

Put ${{\text{x}}^{\text{5}}}=\text{t}$ so that $\text{5}{{\text{x}}^{\text{4}}}\text{dx}=\text{dt}$.

When $\text{x}=-1,\,\text{t}=-1$. When $\text{x}=1,\,\text{t}=1$.

$\text{I}=\int\limits_{\text{-1}}^{\text{1}}{\sqrt{\text{t+1}}\text{dt}}$ 

$=\left[ \dfrac{{{\left( \text{t}+\text{1} \right)}^{\dfrac{3}{2}}}}{\dfrac{3}{2}} \right]_{-1}^{1}=\dfrac{2}{3}\left[ {{\left( \text{t}+\text{1} \right)}^{\dfrac{3}{2}}} \right]_{-1}^{1}$ 

$=\dfrac{2}{3}\left[ {{2}^{\dfrac{3}{2}}}-0 \right]=\dfrac{4\sqrt{2}}{3}$.


Example - 4

Prove that $\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\sqrt{\text{sin}\phi }}\text{co}{{\text{s}}^{\text{5}}}\phi \,\text{d}\phi =\dfrac{\text{64}}{\text{231}}$ 

Ans:

$\text{I}=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\sqrt{\text{sin}\phi }}\text{co}{{\text{s}}^{\text{5}}}\phi \,\text{d}\phi $ 

$=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\sqrt{\text{sin}\phi }}\text{co}{{\text{s}}^{4}}\phi \cos \phi \,\text{d}\phi $ 

$=\int\limits_{\text{0}}^{\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}}{\sqrt{\text{sin}\phi }}{{\left( \text{1}-\text{si}{{\text{n}}^{\text{2}}}\phi  \right)}^{\text{2}}}\text{cos}\phi \,\text{d}\phi $ 

Put $\sin \phi =\text{t}$ so that $\cos \phi \,\text{d}\phi =\text{dt}$.

When $\phi =0,\,\sin 0=\text{t}\Rightarrow \text{t=0}$ 

When $\phi =\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}},\,\text{sin}\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}=\text{t}\Rightarrow \text{t}=\text{1}$ 

$\text{I}=\int\limits_{\text{0}}^{\text{1}}{\sqrt{\text{t}}}{{\left( \text{1}-{{\text{t}}^{\text{2}}} \right)}^{\text{2}}}\text{dt}=\int\limits_{\text{0}}^{\text{1}}{\sqrt{\text{t}}\left( -\text{2}{{\text{t}}^{\text{2}}}+{{\text{t}}^{\text{4}}} \right)\text{dt}}$ 

$=\int\limits_{0}^{1}{\left( {{\text{t}}^{\dfrac{\text{1}}{\text{2}}}}-\text{2}{{\text{t}}^{\dfrac{\text{5}}{\text{2}}}}+{{\text{t}}^{\dfrac{\text{9}}{\text{2}}}} \right)}\text{dt}$ 

$=\left[ \dfrac{{{\text{t}}^{\dfrac{\text{3}}{\text{2}}}}}{\dfrac{\text{3}}{\text{2}}}-\text{2}\dfrac{{{\text{t}}^{\dfrac{\text{7}}{\text{2}}}}}{\dfrac{\text{7}}{\text{2}}}+\dfrac{{{\text{t}}^{\dfrac{\text{11}}{\text{2}}}}}{\dfrac{\text{11}}{\text{2}}} \right]_{\text{0}}^{\text{1}}$ 

$=\left[ \dfrac{\text{2}}{\text{3}}{{\text{t}}^{\dfrac{\text{3}}{\text{2}}}}-\dfrac{\text{4}}{\text{7}}{{\text{t}}^{\dfrac{\text{7}}{\text{2}}}}+\dfrac{\text{2}}{\text{11}}{{\text{t}}^{\dfrac{\text{11}}{\text{2}}}} \right]_{\text{0}}^{\text{1}}$ 

$=\left[ \dfrac{2}{3}\left( 1 \right)-\dfrac{4}{7}\left( 1 \right)+\dfrac{2}{11}\left( 1 \right) \right]-\left[ 0-0+0 \right]$ 

$=\dfrac{2}{3}-\dfrac{4}{7}+\dfrac{2}{11}$ 

$=\dfrac{154-132+42}{231}=\dfrac{64}{231}$ 


Example-5

Evaluate:$\int\limits_{\text{1}}^{\text{2}}{\left( \dfrac{\text{x}-\text{1}}{{{\text{x}}^{\text{2}}}} \right)}{{\text{e}}^{\text{x}}}\text{dx}$ or $\int\limits_{\text{1}}^{\text{2}}{{{\text{e}}^{\text{x}}}\left( \dfrac{\text{1}}{\text{x}}-\dfrac{\text{1}}{{{\text{x}}^{\text{2}}}} \right)}\text{dx}$ 

Ans:

$\int{\left( \dfrac{\text{x}-\text{1}}{{{\text{x}}^{\text{2}}}} \right){{\text{e}}^{\text{x}}}\text{dx}=\int{{{\text{e}}^{\text{x}}}\left( \dfrac{\text{1}}{\text{x}}-\dfrac{\text{1}}{{{\text{x}}^{\text{2}}}} \right)\text{dx}}}$ 

$=\int{\dfrac{\text{1}}{\text{x}}}\text{.}{{\text{e}}^{\text{x}}}\text{dx}-\int{\dfrac{\text{1}}{{{\text{x}}^{\text{2}}}}}\text{.}{{\text{e}}^{\text{x}}}\text{dx}$ 

$=\dfrac{\text{1}}{\text{x}}\text{.}{{\text{e}}^{\text{x}}}-\int{\left( -\dfrac{\text{1}}{{{\text{x}}^{\text{2}}}} \right){{\text{e}}^{\text{x}}}\text{dx}-\int{\dfrac{\text{1}}{{{\text{x}}^{\text{2}}}}\text{.}{{\text{e}}^{\text{x}}}\text{dx}}}$           [Integrating first integral by parts]

$=\dfrac{\text{1}}{\text{x}}\text{.}{{\text{e}}^{\text{x}}}=\text{F}\left( \text{x} \right)$ 

$\int\limits_{\text{1}}^{\text{2}}{\left( \dfrac{\text{x}-\text{1}}{{{\text{x}}^{\text{2}}}} \right){{\text{e}}^{\text{x}}}\text{dx}=\left[ \dfrac{{{\text{e}}^{\text{x}}}}{\text{x}} \right]}_{\text{1}}^{\text{2}}$ 

$=\dfrac{\text{1}}{\text{2}}\text{.}{{\text{e}}^{\text{2}}}-\dfrac{\text{1}}{\text{1}}{{\text{e}}^{\text{1}}}=\dfrac{\text{1}}{\text{2}}{{\text{e}}^{\text{2}}}-\text{e}$.


Area Under the Curves

Example - 1

Find the area bounded by the curve $\text{y}={{\text{x}}^{\text{2}}}-\text{5x}+\text{6}$, $\text{X-}$axis and the lines $\text{x}=\text{1}$ and 4.

Ans:

For $\text{y}=\text{0}$, we get ${{\text{x}}^{\text{2}}}+5\text{x}+6=0\Rightarrow \text{x}=2,3$ 

Hence the curve crosses $\text{X-}$axis at $\text{x}=\text{2,3}$ in the interval $\left[ 1,4 \right]$.

Bounded Area$\text{=}\left| \int\limits_{\text{1}}^{\text{2}}{\text{ydx}} \right|\text{+}\left| \int\limits_{\text{2}}^{\text{3}}{\text{ydx}} \right|\text{+}\left| \int\limits_{\text{3}}^{\text{4}}{\text{ydx}} \right|$ 


Area under the curves

$\Rightarrow \text{A=}\left| \int\limits_{\text{1}}^{\text{2}}{\left( {{\text{x}}^{\text{2}}}-\text{5x+6} \right)\text{dx}} \right|\text{+}\left| \int\limits_{\text{2}}^{\text{3}}{\left( {{\text{x}}^{\text{2}}}-\text{5x+6} \right)\text{dx}} \right|\text{+}\left| \int\limits_{\text{3}}^{\text{4}}{\left( {{\text{x}}^{\text{2}}}-\text{5x}+\text{6} \right)\text{dx}} \right|$ 

${{\text{A}}_{\text{1}}}=\left[ \dfrac{{{2}^{3}}-{{1}^{3}}}{3} \right]-5\left( \dfrac{{{2}^{2}}-{{1}^{2}}}{2} \right)+\left[ 6\left( 2-1 \right) \right]=\dfrac{5}{6}$ 

${{\text{A}}_{\text{2}}}=\dfrac{{{3}^{3}}-{{2}^{3}}}{3}-5\left( \dfrac{{{3}^{2}}-{{2}^{2}}}{2} \right)+6\left( 3-2 \right)=-\dfrac{1}{6}$ 

${{\text{A}}_{\text{3}}}=\dfrac{{{4}^{3}}-{{3}^{3}}}{3}-5\left( \dfrac{{{4}^{2}}-{{3}^{2}}}{2} \right)+6\left( 4-3 \right)=\dfrac{5}{6}$ 

$\Rightarrow \text{A}=\dfrac{5}{6}+\left| -\dfrac{1}{6} \right|+\dfrac{5}{6}=\dfrac{11}{6}\,\text{sq}\text{.}\,\text{units}$ 


Example - 2 

Find the area bounded by the curve: $\text{y}=\sqrt{\text{4}-\text{x}}\text{,}$ $\text{X-}$axis and $\text{Y-}$axis.

Ans:

Trace the curve $\text{y}=\sqrt{\text{4}-\text{x}}$.

  1.  Put $\text{y}=\text{0}$ in the given curve to get $\text{x}=\text{4}$ as the point of intersection with X-axis.

Put $\text{x}=\text{0}$ in the given curve to get $\text{y}=2$ as the point of intersection with Y-axis.

  1. For the curve, $\text{y}=\sqrt{\text{4}-\text{x}}\text{,}\,\text{4}-\text{x}\ge \text{0}$ 

$\Rightarrow \text{x}\le \text{4}$ 

$\Rightarrow $ curve lies only to the left of $\text{x}=4$ line.

  1. As any $\text{y}$ is positive, curve is above X-axis.


Area under the given curve is above x-axis

Using step $1$ to $3$, we can draw the rough sketch of $\text{y}=\sqrt{\text{4}-\text{x}}$ 

In figure, Bounded area$=\int\limits_{\text{0}}^{\text{4}}{\sqrt{\text{4}-\text{x}}\text{dx}=\left| \dfrac{-\text{2}}{\text{3}}\left( \text{4}-\text{x} \right)\sqrt{\text{4}-\text{x}} \right|}_{\text{0}}^{\text{4}}=\dfrac{\text{16}}{\text{3}}\,\text{sq}\text{.}\,\text{units}$ 


Example - 3

$\text{AOBA}$ is the part of the ellipse $\text{9}{{\text{x}}^{\text{2}}}+{{\text{y}}^{\text{2}}}=\text{36}$ in the first quadrant such that $\text{OA}=\text{2}$ and \[\text{OB}=6\]. Find the area between the arc $\text{AB}$ and the chord $\text{AB}$.

Ans:

The given equation of the ellipse can be written as 

$\dfrac{{{\text{x}}^{\text{2}}}}{\text{4}}+\dfrac{{{\text{y}}^{\text{2}}}}{\text{36}}=1\,\text{i}\text{.e}\text{.}\,\dfrac{{{\text{x}}^{\text{2}}}}{{{\text{2}}^{\text{2}}}}+\dfrac{{{\text{y}}^{\text{2}}}}{{{\text{6}}^{\text{2}}}}=1$ 


Area of the required region in ellipse

$\text{A}$ is $\left( 2,0 \right)$ and $\text{B}$ is $\left( 0,6 \right)$.

The equation of chord $\text{AB}$ is:

$\text{y}-0=\dfrac{6-0}{0-2}\left( \text{x}-\text{2} \right)$ 

$\Rightarrow \text{y}=-3\text{x}+6$.

Reqd. area (shown shaded)

$=\int\limits_{0}^{2}{3\sqrt{4-{{\text{x}}^{\text{2}}}}\text{dx}-\int\limits_{0}^{2}{\left( 6-3\text{x} \right)}}\text{dx}$ 

$=\text{3}\left[ \dfrac{\text{x}\sqrt{\text{4}-{{\text{x}}^{\text{2}}}}}{\text{2}}+\dfrac{\text{4}}{\text{2}}\text{si}{{\text{n}}^{-\text{1}}}\dfrac{\text{x}}{\text{2}} \right]_{\text{0}}^{\text{2}}-\left[ \text{6x}-\dfrac{\text{3}{{\text{x}}^{\text{2}}}}{\text{2}} \right]_{\text{0}}^{\text{2}}$ 

$=3\left[ \dfrac{2}{2}\left( 0 \right)+2{{\sin }^{-1}}\left( 1 \right) \right]-\left[ 6\left( 2 \right)-\dfrac{3\left( 4 \right)}{2} \right]$ 

$=3\left[ 2\times \dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}} \right]-\left[ 12-6 \right]$ 

$=\left( \text{3 }\!\!\pi\!\!\text{ }-\text{6} \right)\,\text{sq}\text{.}\,\text{units}$ 


Example - 4

Find the area bounded by the curves $\text{y}={{\text{x}}^{\text{2}}}$ and ${{\text{x}}^{\text{2}}}+{{\text{y}}^{\text{2}}}=2$ above X-axis.

Ans:

Let us first find the points of intersection of curves.

Solving $\text{y}={{\text{x}}^{\text{2}}}$ and ${{\text{x}}^{\text{2}}}+{{\text{y}}^{\text{2}}}=2$ simultaneously, we get:

${{\text{x}}^{\text{2}}}+{{\text{x}}^{\text{4}}}=2$ 

$\Rightarrow \left( {{\text{x}}^{\text{2}}}-1 \right)\left( {{\text{x}}^{\text{2}}}+2 \right)=0$ 

$\Rightarrow {{\text{x}}^{\text{2}}}=\text{1}\,\,\text{and}\,\,{{\text{x}}^{\text{2}}}=-\text{2}$ [reject]

$\Rightarrow \text{x}=\pm 1$ 

$\Rightarrow \text{A}=\left( -\text{1,1} \right)$ and $\text{B}=\left( 1,1 \right)$ 

Shaded Area$=\int\limits_{-\text{1}}^{\text{+1}}{\left( \sqrt{\text{2}-{{\text{x}}^{\text{2}}}}-{{\text{x}}^{\text{2}}} \right)\text{dx}}$ 

$=\int\limits_{-1}^{+1}{\sqrt{\text{2}-{{\text{x}}^{\text{2}}}}\text{dx}-\int\limits_{\text{-1}}^{\text{+1}}{{{\text{x}}^{\text{2}}}\text{dx}}}$ 

$=2\int\limits_{0}^{1}{\sqrt{\text{2}-{{\text{x}}^{\text{2}}}}\text{dx}-2\int\limits_{\text{0}}^{\text{1}}{{{\text{x}}^{\text{2}}}\text{dx}}}$ 

$=\text{2}\left[ \dfrac{\text{x}}{\text{2}}\sqrt{\text{2}-{{\text{x}}^{\text{2}}}}\text{+}\dfrac{\text{2}}{\text{2}}\text{si}{{\text{n}}^{-\text{1}}}\dfrac{\text{x}}{\sqrt{\text{2}}} \right]_{\text{0}}^{\text{1}}-\text{2}\left( \dfrac{\text{1}}{\text{3}} \right)$ 

$=\text{2}\left( \dfrac{\text{1}}{\text{2}}+\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{4}} \right)-\dfrac{\text{2}}{\text{3}}=\dfrac{\text{1}}{\text{3}}\text{+}\dfrac{\text{ }\!\!\pi\!\!\text{ }}{\text{2}}\,\,\text{sq}\text{.}\,\,\text{units}$ 


Area of the bounded region above x-axis

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